16.6 Expected Value

Sometimes rather than wanting to find the probability of one event happening, you want to know what will most likely happen in a given situation. The average value, or the expected value, is the long-term average of the probability of an event. The expected value is the sum of the probability of each possible outcome of an event multiplied by the value of each outcome.

Payoff x1 x2 x3 … xn Probability P(x1) P(x2) P(x3) P(xn) If a given situation involves various payoffs then its expected values is calculated as follows. Payoff x1 x2 x3 … xn Probability P(x1) P(x2) P(x3) P(xn)

-$6 The Spinner Game = -0.5 lose $.50 The house has the advantage. In the spinner game you get to spin the spinner shown and win or lose the amount that you land on. To find the expected value that you will win, you use the same expected value formula:   Expected Value: You would most likely ____________ if you played this game. -$6   = -0.5 lose $.50 The house has the advantage.

Dice Game – you and your friend make up the following game. Event Die shows 1,2, or 3 Die shows 4 or 5 Die shows 6 Payoff +10 pts. -13 pts. -1 pt. Probability 1/2 1/3 1/6 What is the expected value? 10(1/2) – 13(1/3) – 1(1/6) Win .5 points You would most likely ______________ if you played this game. You would have the advantage.

FAIR GAME?? IF the expected value of a game is ZERO, then the game is called a fair game. The dice game and spinner game were not fair games.

0 game is fair 4(1/4) – 1(1/2) – 2(1/4) Heads 2 1 Payoff to A Two coins are tossed. If both land heads up, then player A wins $4 from player B. If exactly one coin lands heads up, then B wins $1 from A. If both land tails up then B wins $2 from A . Is this a fair game? Heads 2 1 Payoff to A Probability -$2 $4 -$1 1/4 1/2 1/4 What is the expected value of the game? 4(1/4) – 1(1/2) – 2(1/4) 0 game is fair

2(15/36) – 1(21/36) $0.25, game is not fair The Payout game – If the sum of two rolled dice is 8 or more, you win $2.00; if not you lose $1.00. Is this a fair game? Event Sum ≥ 8 Sum < 8 Payoff $2 -$1 Probability 15/36 21/36 What is the expected value of the game? 2(15/36) – 1(21/36) $0.25, game is not fair

What payoff value(s) would make the game fair? Event Sum ≥ 8 Sum < 8 Payoff $2 -$1 Probability 15/36 21/36 X(15/36) – 1(21/36) = 0 OR 2(15/36) – X(21/36) = 0 $1.40 -$1.43

Three cards are drawn at random without replacement, from a standard deck. Find the expected value for the occurrence of hearts. Hearts 1 2 3 Probability 703/1700 741/1700 117/850 11/850   =.75

Homework p.633-635 #1, 5-7,11,13,14, 22